Sheath Theory Research Articles

Correlation between non-ambipolar currents and divertor heat loads in the COMPASS tokamak

Abstract Electric current flowing onto the divertor of the COMPASS tokamak influences its heat loading. The current measured when a Langmuir probe is grounded to the divertor gives a local measurement of this heat loading, which, according to the classical theory of the Debye sheath, should be enhanced with respect to the case of locally ambipolar currents. The comparison of the calculated heat flux by probes with infrared thermography, when the influence of non-ambipolar currents is not considered, is grossly wrong; when the theoretical effect of non-ambipolar currents is included, however, the agreement is very good.

Self-Coagulation Theory and Related Comet- and Semi-Circle-Shaped Structures in Electronegative and Gaseous Discharging Plasmas in the Laboratory

In this work, the two-dimensional fluid models for two types of inductively coupled plasma, Ar/O2 and Ar/SF6, are numerically solved by the finite element method. Four interesting phenomena revealed by the simulations are reported: (1) comet-shaped and semi-circle-shaped structures in Ar/O2 and Ar/SF6 plasmas, respectively; (2) blue sheaths that surround the two structures; (3) the collapse and dispersion of semi-circle-shaped structures of certain Ar/SF6 plasma cations and anions when they are observed separately; and (4) the rebuilding of coagulated structures by minor cations in the Ar/SF6 plasma at the discharge center. From the simulation detail, it was found that the cooperation of free diffusion and negative chemical sources creates the coagulated structure of anions, and the self-coagulation theory is therefore built. The advective and ambipolar types of self-coagulation are put forth to explain the co-existence of blue sheath and internal neutral plasma, among which the advective type of self-coagulation extends the Bohm’s sheath theory of cations to anions, and the ambipolar type of self-coagulation originates from the idea of the ambipolar diffusion process, and it updates the recognition of people about the plasma collective interaction. During the ambipolar self-coagulation, each type of Ar/SF6 plasma cations and anions is self-coagulated, and the coagulated plasma species are then modeled as mass-point type (or point-charge type, more precisely). When the charge amounts of two point-charge models of plasma species with the same charge type are equal, the expelling effect caused by the Coulomb’s force of them leads to the collapse or dispersal of heavily coagulated species. The simulation shows that the lighter the species is, the easier it self-coagulates and the more difficult its coagulation is broken, which implies the inertia effect of density quantity. Moreover, the collapse of cation coagulation creates the spatially dispersed charge cloud that is not shielded into the Debye’s length, which indicates the anti-collective behavior of electronegative plasmas when they are self-coagulated. The rebuilt coagulated structure of minor Ar/SF6 plasma species at the discharge center and the weak coagulation of electrons in the periphery of the main coagulated structure that is under the coil are caused by the monopolar and spontaneous (non-advective) type of self-coagulation. The analysis predicts an intensity order of physically driven coagulation force, chemical self-coagulation force, and ambipolar self-coagulation force. The popular coagulated structure of the electronegative ICP sources is urgently needed to validate the experiment.

Non-Maxwellian electron effects on the macroscopic response of a Hall thruster discharge from an axial–radial kinetic model

A 2D axial–radial particle-in-cell (PIC) model of a Hall thruster discharge has been developed to analyze (mainly) the fluid equations satisfied by the azimuthally-averaged slow dynamics of electrons. Their weak collisionality together with a strong interaction with the thruster walls lead to a non-Maxwellian velocity distribution function (VDF). Consequently, the resulting macroscopic response differs from a conventional collisional fluid. First, the gyrotropic (diagonal) part of the pressure tensor is anisotropic. Second, its gyroviscous part, although small, is relevant in the azimuthal momentum balance, where the dominant contributions are orders of magnitude lower than in the axial momentum balance. Third, the heat flux vector does not satisfy simple laws, although convective and conductive behaviors can be identified for the parallel and perpendicular components, respectively. And fourth, the electron wall interaction parameters can differ largely from the classical sheath theory, based on near Maxwellian VDF. Furthermore, these effects behave differently in the near-anode and near-exit regions of the channel. Still, the profiles of basic plasma magnitudes agree well with those of 1D axial fluid models. To facilitate the interpretation of the plasma response, a quasiplanar geometry, a purely-radial magnetic field, and a simple empirical model of cross-field transport were used; but realistic configurations and a more elaborate anomalous diffusion formulation can be incorporated. Computational time was controlled by using an augmented vacuum permittivity and a stationary depletion law for neutrals.

Two-dimensional collisional particle model of the divertor sheath with electron emissive walls

A novel two-dimensional particle-in-cell (PIC) code, named Divertor Edge Simulator of Plasma-wall Interaction with Consistent COllisions (DESPICCO) and developed at CNR-ISTP, is capable of simulating the thin plasma layer of several millimeters, adjacent to the divertor tiles of a Tokamak fusion reactor. Here, kinetic effects and non-neutral plasma physics in the Debye sheath can be self-consistently captured by the PIC approach. The code is firstly benchmarked against literature one-dimensional codes and additional theoretical predictions for a magnetized sheath. Then, it is applied to a realistic divertor scenario featuring an attached plasma with monoblocks (MBs) radial misalignment and gaps, to compute the energy flux amplification factor at the exposed MB edge. A non-ambipolar local current density close to the leading edge and an average sheath heat transmission coefficient larger than the one predicted by classical sheath theory, are found. The effects of electron wall emission and plasma-gas collisions on the ion Mach number and on particle and energy fluxes to the walls are finally estimated to determine future guidelines for simulations. Ion collisions with recycled neutrals and both secondary and thermionic electron emission from the wall are found to have a relevant impact, with the overall effect of reducing by 25% the average ion impact energy, and by 15%–20% the total heavy particles energy flux to the walls, with relevant implications on the divertor wall erosion.

New achievement of the global sheath-bulk model for the collisionless radio-frequency using in scale industries

Plasma processing is extensively utilized in enormous industries products like semiconductor devices, textile fabrics, modifying polymers, and seed treatments. One of the most significant enhancements to this technology is the low pressure of capacitively coupled plasma (CCPs). Contrary to other plasma processes like atmospheric pressure plasma jet, low-pressure plasma under vacuum is the best choice for scale industrial applications, including etching, semiconductor IC fabrication, and seed processing for agricultural use. This is due to its full soft control and complete ionization chamber. It is quite difficult to analyze the dynamics of the plasma sheath theory. Despite being among the most relevant models, the step approximation model does not fully account for all nonlinear dynamics, particularly high harmonic effects, the particle density singularity between the sheath and bulk regions, and the absence of an ambipolar field in the bulk zone. We provide a significant solution to this issue in this study. The numerical fluid model for collisionless self-consistency is solved. As a result, the sheath charge distribution V(Q), which is considered as the main factor of control all nonlinear dynamics of the sheath, is estimated. Strong agreement between experimental data from the Ziegler et al. [7] study and our cubic fitting formula for the V(Q) distribution. Additionally, by compared the results of step approximation with the non step model, theoretical advancements like handling the particle density singularity and the merging of the sheath-bulk zone are further illustrated. Moreover, the most simple but effective solution at a minimal computing cost, the global model, is solved. By applying accurate sheath charge distribution, which controls all nonlinear dynamics, the global model's accuracy is established. As a result, more nonlinear dynamics are achieved, including distributions of currents, densities, charges, and power. Finally, excellent calculations of average power and sheath distance in case of collisionless region are calculated. Our study can find a more accurate physical and engineering approach which may help in industry applications.

Vlasov simulation of the emissive plasma sheath with energy-dependent secondary emission coefficient and improved modeling for dielectric charging effects

A one-dimensional Vlasov–Poisson simulation code is employed to investigate the plasma sheath considering electron-induced secondary electron emission (SEE) and backscattering. The SEE coefficient is commonly treated as constant in a range of plasma simulations; here, an improved SEE model of a charged dielectric wall is constructed, which includes the wall charging effect on the SEE coefficient and the energy dependency of the SEE coefficient. Pertinent algorithms to implement the previously mentioned SEE model in plasma simulation are studied in detail. It is found that the SEE coefficient increases with the amount of negative wall charges, which in turn reduces the emissive sheath potential. With an energy-dependent SEE coefficient, the sheath potential is a nonlinear function of the plasma electron temperature, as opposed to the linear relation predicted by the classic emissive sheath theory. Simulation combining both wall-charging effect and SEE coefficient’ energy dependency suggests that the space-charged limited sheath is formed at high plasma electron temperature levels, where both sheath potential and surface charging saturate. Additionally, different algorithms to implement the backscattering in the kinetic simulation are tested and compared. Converting backscattered electrons to secondary electrons via an effective SEE coefficient barely affects the sheath properties. The simulation results are shown to be commensurate with the upgraded sheath theory predictions.

On the electron sheath theory and its applications in plasma–surface interactions

In this work, an improved understanding of electron sheath theory is provided using both fluid and kinetic approaches while elaborating on their implications for plasma–surface interactions. A fluid model is proposed considering the electron presheath structure, avoiding the singularity in electron sheath Child–Langmuir law which overestimates the sheath potential. Subsequently, a kinetic model of electron sheath is established, showing considerably different sheath profiles in respect to the fluid model due to non-Maxwellian electron velocity distribution function and finite ion temperature. The kinetic model is then further generalized and involves a more realistic truncated ion velocity distribution function. It is demonstrated that such a distribution function yields a super-thermal electron sheath whose entering velocity at the sheath edge is greater than the Bohm criterion prediction. Furthermore, an attempt is made to describe the electron presheath–sheath coupling within the kinetic framework, showing a necessary compromise between a realistic sheath entrance and the inclusion of kinetic effects. Finally, the secondary electron emissions induced by sheath-accelerated plasma electrons in an electron sheath are analysed and the influence of backscattering is discussed.

Comprehensive kinetic theory of inverse sheath for a strong electron-emitting electrode in a low-pressure isotropic plasma

The basic kinetic theory of an electron emitting inverse sheath was presented in T Gyergyek, J Kovačič, I Gomez, J P Gunn, S Costea and M Mozetič (2020 Phys. of Plasmas 27, 023 520). Here we extend this theory to find the potential profile and kinetic energy flux in inverse sheath for floating and current carrying electron emitting electrode/wall. The values of emitted-electron temperature, number and current densities are explored for the existence and nonexistence of inverse sheath for floating and current carrying electrode/wall. For this we consider half Maxwellian velocity distribution functions of species (emitted-electron, plasma-electron and ions) at their respective emerging boundaries. The species charge densities are calculated self-consistently from the prior assumed positive sheath structure. The Poisson’s equation is then solved numerically for floating and current carrying electrode/wall with varying normalized emitted-electron and ion temperatures. The resulting inverse sheath solution is valid for limited range of emitted-electron and ion temperatures in case of floating electrode/wall. The kinetic energy flux relations for each species are derived in inverse sheath. The numerical solutions of these relations for floating and current carrying electrode/wall are presented for valid range of parameters. These solutions shows that the total or kinetic flux received by floating electrode/wall surface decreases with increasing of emitted-electron temperature and even approaches to zero for equal values of emitted-electron and plasma-electron temperatures.

Influence of the Ion Mass in the Radial to Orbital Transition in Weakly Collisional Low-Pressure Plasmas Using Cylindrical Langmuir Probes

This paper presents an experimentally observed transition from the validity of the radial theories to the validity of the orbital theories that model the ion current collected by a cylindrical Langmuir probe immersed in low-pressure, low-temperature helium plasma when it is negatively biased with respect to the plasma potential, as a function of the positive ion-neutral collision mean free path to the Debye length ratio Λ=λ+/λD. The study has been also conducted on argon and neon plasmas, which allows a comparison based on the mass of the ions, although no transition has been observed for these gases. As the radial or orbital behavior of the ions is essential to establish the validity of the different sheath theories, a theoretical analysis of such a transition not only as a function of the parameters Λ and β=T+/Te, T+ and Te being the positive ion and electron temperature, respectively, but also as a function of the ion mass is provided. This study allows us to recognize the importance of the mass of the ion as the parameter that explains the transition in helium plasmas. Motivated by these theoretical arguments, a novel set of measurements has been performed to study the relationship between the Λ and β parameters in the transition that demonstrate that the effect of the ion mean free path cannot be completely ignored and also that its influence on the ion current collected by the probe is less important than the effect of the ion temperature.

Deuterium experiment with large-scale negative ion source for large helical device

The deuterium experiment with various source plasma diagnostics was conducted at the research and development negative ion source at National Institute for Fusion Science, which is a cesium-seeded large-scale negative hydrogen-ion source with half discharge volume and the same type of ion source for large helical device. A change of filling gas from hydrogen to deuterium increased line-averaged negative ion density by a factor of 1.3 in beam extraction region as a result of flattened profile with keeping central density. Positive ion and electron densities increased threefold. This was influenced by the increase of plasma density in the plasma generation region. The plasma space potential varied as equivalent of the collisionless sheath theory in electro-positive plasma expected by the hydrogen isotope gas species, although the negative ion could affect sheath formation. Higher atomic cesium density was observed because of higher sputtering yield due to higher momentum of deuterium.

Experimental investigation of sheath effects on I–V traces of strongly electron emitting probes

I–V traces of strongly emitting emissive probes are investigated in a multidiople filament discharge. It is found that at sufficiently high neutral pressure and emitting current, the variation of the I–V traces and their associated inflection points no longer follow the previous predictions of space charge limited (SCL) models. A new, steep slope region of the I–V trace appears near the plasma potential when the probe is strongly emitting, causing the inflection point and the floating potential to increase towards the plasma potential as emission current increases, rather than staying constant. This is, to our knowledge, the first experimental evidence that the effects predicted by Campanell et al’s inverse sheath theory (2017 Physics of Plasmas 24 057101) not only affect the floating potential but also a region in the I–V trace of an emissive probe. It is also found that the double inflection point structure when the probe is biased below the ionization energy of the working gas is highly likely to be an emission retardation effect from enhanced virtual cathode formation due to the increased local electron density. The implications of these findings on hot cathode sources are briefly discussed.

An experimental study of a dual-stage 4-grid ion thruster

Performance experiments of the dual-stage 4-grid ion thruster were conducted. Experimental results showed that the specific impulse reached 7100 s and the beam divergence angle reached 6.3° when the screen and extraction potentials were 4.5 and 2.5 kV, respectively. The test results of beam extraction characteristics showed that the extraction capacity of ions was determined by extraction potential and was independent of screen grid and acceleration stage potentials. The extraction capacity of ions was analyzed with plasma sheath theory, and conclusions were in accordance with the experimental results. This study presents the decoupling mechanism of ion extraction and acceleration processes of the dual-stage ion 4-grid optical system by experiment and simulation.

Kinetic Theory of the Wall Sheath for Arbitrary Conditions in a Gas-Discharge Plasma

The self-consistent problem of the structure of a perturbed wall sheath in a dc gas-discharge plasma near a flat surface under negative potential relative to the plasma has been solved for an arbitrary relation between the Debye radius and the ion mean free path. The solution has been obtained without artificial separation of this layer into the quasi-neutral “presheath” and the wall sheath in which quasi-neutrality is violated substantially. The actual ion distribution function in the unperturbed plasma, the dependence of the charge-exchange cross section on the ion energy, and the nonzero electric field in the unperturbed plasma have been considered. It is shown that when the average electron energy is conserved the structure of the perturbed wall sheath weakly depends on the form of the electron distribution function. It has been established that the mean energy of ions in the unperturbed plasma substantially affects the structure of the quasi-neutral presheath as well as the structure of a part of the wall sheath in which quasi-neutrality is not observed even under the assumption that the mean electron energy is much higher than the mean energy of ions. The calculations of ion flow parameters and the structure of the perturbed wall sheath are in conformity with experimental data obtained by other authors, which could not be adequately interpreted earlier.

Lunar surface potential and electric field**A contributed paper from the International Symposium on Lunar and Planetary Science (ISLPS) on 2018 June 12–15 at Macau University of Science and Technology.

The Moon has no significant atmosphere, thus its surface is exposed to solar ultraviolet radiation and the solar wind. Photoemission and collection of the solar wind electrons and ions may result in lunar surface charging. On the dayside, the surface potential is mainly determined by photoelectrons, modulated by the solar wind; while the nightside surface potential is a function of the plasma distribution in the lunar wake. Taking the plasma observations in the lunar environment as inputs, the global potential distribution is calculated according to the plasma sheath theory, assuming Maxwellian distributions for the surface emitted photoelectrons and the solar wind electrons. Results show that the lunar surface potential and sheath scale length change versus the solar zenith angle, which implies that the electric field has a horizontal component in addition to the vertical one. By differentiating the potential vertically and horizontally, we obtain the global electric field. It is found that the vertical electric field component is strongest at the subsolar point, which has a magnitude of 1 V m−1. The horizontal component is much weaker, and mainly appears near the terminator and on the nightside, with a magnitude of several mV m−1. The horizontal electric field component on the nightside is rotationally symmetric around the wake axis and is strongly determined by the plasma parameters in the lunar wake.

Test cases for grid-based direct kinetic modeling of plasma flows

Grid-based kinetic models are promising in that the numerical noise inherent in particle-based methods is essentially eliminated. Here, we call such grid-based techniques a direct kinetic (DK) model. Velocity distribution functions are directly obtained by solving kinetic equations, such as the Vlasov equation, in discretized phase space, i.e., both physical and velocity space. In solving the kinetic equations that are hyperbolic partial differential equations, we employ a conservative, positivity-preserving numerical scheme, which is necessary for robust calculations of problems particularly including ionization. Test cases described in this paper include plasma sheaths with electron emission and injection and expansion of neutral atom flow in a two-dimensional configuration. A unifying kinetic theory of space charge limited sheaths for both floating and conducting surfaces is presented. The improved theory is verified using the collisionless DK simulation, particularly for small sheath potentials that particle-based kinetic simulations may struggle due to statistical noise. For benchmarking of the grid-based and particle-based kinetic simulations, hybrid simulations of Hall thruster discharge plasma are performed. While numerical diffusion occurs in the phase space in the DK simulation, ionization oscillations are well resolved since ionization events can be taken into account deterministically at every time step.

Compensation of the sheath effects in cylindrical floating probes

In cylindrical floating probe measurements, the plasma density and electron temperature are overestimated due to sheath expansion and oscillation. To reduce these sheath effects, a compensation method based on well-developed floating sheath theories is proposed and applied to the floating harmonic method. The iterative calculation of the Allen-Boyd-Reynolds equation can derive the floating sheath thickness, which can be used to calculate the effective ion collection area; in this way, an accurate ion density is obtained. The Child-Langmuir law is used to calculate the ion harmonic currents caused by sheath oscillation of the alternating-voltage-biased probe tip. Accurate plasma parameters can be obtained by subtracting these ion harmonic currents from the total measured harmonic currents. Herein, the measurement principles and compensation method are discussed in detail and an experimental demonstration is presented.

Probing the electron density in HiPIMS plasmas by target inserts

High power impulse magnetron sputtering (HiPIMS) is a versatile technology to deposit thin films with superior properties. During HiPIMS, the power is applied in short pulses of the order of 100 μs at power densities of kW to a magnetron target creating a torus shaped dynamic high density plasma. This plasma torus is not homogeneous, but individual ionization zones become visible, which rotate along the torus with velocities of 10 km . Up to now, however, any direct measurement of the electron density inside these rotating ionization zones is missing. Here, we probe the electron density by measuring the target current locally by using small inserts embedded in an aluminium target facing the plasma torus. By applying simple sheath theory, a plasma density of the order of at the sheath edge can be inferred. The plasma density increases with increasing target current. In addition, the dynamics of the local target current variation is consistent with the dynamics of the traveling ionization zone causing a modulation of the local current density by 25%.

Three-dimensional direct numerical simulation of helicon discharge

With the detailed consideration of electrochemical reactions and collision relations, a direct numerical simulation model of helicon plasma discharge with three-dimensional fluid-dynamic equations is proposed in the present work. It can improve the precision of results and widen the model applicability by discarding the small perturbation theory in previous helicon models which are partially analytical in essence. Under the assumption of weak ionization, the Maxwell equations coupled with the plasma parameters are directly solved in the whole computational domain. Thus the energy deposited from electromagnetic wave to plasma can be then easily calculated. The values of plasma parameters which include electron density, mean electron energy and heavy species density are obtained by solving a set of drift-diffusion equations. Meanwhile, seven kinds of chemical reactions in the plasma and two kinds of surface reactions on the wall are taken into account. All of the partial differential equations are solved by the finite element solver of COMSOL MultiphysicsTM with the full coupled method.#br#The results of numerical cases employing argon as the working medium show that there exists a sharp density jump from a low to high value as the radiofrequency power is raised. The density jump phenomenon is in accordance with the experimental results of Toki (Toki K, Shinohara S, Tanikawa T, Shamrai K P 2006 Thin Solid Films 506-507 597) and Chen (Chen F F 2007 Plasma Sources Sci. Technol. 16 593). The electron temperature decreases with an increase of the gas pressure, which is similar to Toki's (Toki K, Shinohara S, Tanikawa T, Shamrai K P 2006 Thin Solid Films 506-507 597) measurement by a RF compensation probe. In comparison with the classical sheath theory, the simulation result demonstrates that the distribution of parameters such as particle number density, the Deby length, electric potential and electron temperature can be solved exactly. In addition, the phenomenon of low-field density peak in helicon discharge was studied in the work. Previous research by Chen (Chen F F 2003 Phys. Plasmas 10 2586) suggests that this peak is caused by constructive interference from the reflected wave. The effect of length of the discharge chamber on the relation of electron density and background magnetic field is investigated numerically. The results validate the mechanism of wave interference reflected by endplates of the discharge chamber. Furthermore, the time-averaged magnetic energy density has more than one peak on the axial direction. Comparing the distribution of the magnetic energy density to that of the dimensionless amplitude of the helicon wave and the TG wave in the one-dimensional undamped condition, it found that the length of peak to peak of the helicon wave is just as twice as that of the magnetic energy density, which indicates that the substance of wave interference is involved in the standing wave generated by the helicon wave and its reflected wave from endplates.

The characteristic of magnetized plasma sheath with nanometer-sized dust grains in fusion devices

It is well known that dust grains may affect the sheath structure when dust grains are dense. Typical fluid theory of a dusty plasma sheath assumes that the charging relaxation time of dust grains is much smaller than the time of dust grains across the sheath and the orbit motion limited theory is used to calculate the dust charge. Under these conditions, it is shown that dust charge in the sheath and the sheath parameters associated with dust charge fluctuate within a certain range. However, for fusion plasma, dust grains with large velocity and small radius can often be observed in the edge region of fusion devices. In this case, charge relaxation time may be much longer than the transit time moving from the sheath edge to the wall and then the dust charge can be considered constant. In this paper, one-dimensional multifluid model is developed to study a sheath of dusty plasma containing nanometer-sized dust grains with constant charge based on their long charging relaxation time and the short time across the sheath. It is found that the sheath parameters show monotonic variations and differ from the oscillation variations obtained from the sheath model which the charging relaxation time of dust grains is much smaller than the time across the sheath. In addition, the effects of dust grain density at the sheath edge, dust radius, and ion Mach number on sheath parameters are studied.

Dust levitation and transport over the surface of the Moon

Exposed to space plasma and solar radiation, electrostatic potential may build up in the lunar regolith, leading to a wealth of dust phenomena, including levitation, oscillation, and transport over the surface. Based on plasma sheath theory, the global near-surface plasma environment is modeled, and the dynamics of charged dust are investigated. Results show that sub-micron sized dust particles can be levitated by the electric field over the surface, forming a dust belt that changes in position and thickness depending on the solar zenith angle. On the dayside of the Moon, stably levitated particles are about ten times smaller, and collect in a thinner belt closer to the surface than do those on the nightside. Although the size and charge of stably levitated dust particles are dependent on ambient plasma conditions, initial charge and velocity, which are closely related to the dynamics of dust particles including charging, oscillation, and damping, will determine whether, or not, a particle can attain stable levitation. Horizontal electrostatic dust transport near to the terminator region may lead to net deposition of dust from the dark into the sunlit hemisphere. Finally, because of different charging processes that result due to rotation of the Moon, before precipitation, dust particles in the dusk terminator region may be transported much longer distances and oscillate to much higher altitude than these in the dawn terminator.